1. Field of the Invention
This invention relates generally to the field of seismic data processing. Specifically, the invention is a method for analyzing spatially-varying noise in seismic data.
2. Description of the Related Art
Seismic data, like virtually any data, are typically noisy. The noise in seismic data affects the accuracy and reliability of products and interpretations derived from the data. Noise removal methods only provide, in general, an estimate of the true value of the underlying signal, and therefore are not completely adequate. More specifically, such estimates have uncertainties that current noise removal methods cannot adequately characterize, and those uncertainties propagate into all subsequent uses of the data. Therefore, there is a need to identify the noise in seismic data, remove it as best as possible, and quantify the uncertainty in the signal that remains.
Seismic data derived from two-dimensional or three-dimensional surveys do not have constant signal or noise levels across the entire geographical extent of the surveys. The term “seismic data” as used here includes electro-seismic data. Varying illumination, statics, wave inferences, and acquisition geometry, among other parameters, affect to varying degrees the reliability of data at different locations in the subsurface. As a result, the spatial variability of the noise inherent to a survey cannot be fully modeled before the acquisition of the data. There is a need for a method to identify and quantify the noise in the data based on analysis of information from the data themselves, and to do this in a manner that accommodates the spatial variability in data quality.
Because of the presence of noise, the properties of the subsurface deduced from survey data have uncertainties. These uncertainties limit the petroleum industry's ability to characterize the nature of hydrocarbon reservoirs, and thereby constrain decisions on reservoir economics and development strategies. Nonetheless, if alternative scenarios for the reservoir's properties can be generated that are consistent with the observed data, then those alternatives can be used to explore the corresponding reservoir economics and development strategies. Through this approach, one may identify the strategies that, given the uncertainty, will most likely achieve the desired goals. This process of generating multiple alternative scenarios, or multiple “realizations,” is an important aspect of petroleum reservoir development, exploitation, and financial planning.
Several approaches have been proposed to address noise identification, noise removal, and uncertainty quantification in seismic data. These include geostatistics, trend surface analysis, signal processing and filtering, and Markov chain analysis.
Geostatistical methods involve linear interpolation to a specific geographic location given data at other geographic locations. Geostatistical methods are particularly useful in the modeling of sparse data, and have been shown to be a specific type of spline fitting. It is understood that such techniques make the assumption that the data are error free, an assumption that is inappropriate for modeling noisy seismic data. More specifically, as geostatistical realizations enumerate the spline paths that may possibly connect the data points, they do not address whether the data underlying the interpolations are reliable.
Approaches have been proposed to incorporate seismic observations into geostatistical models, for example by use of a non-statistical method that involves annealing in numerical aspects of the seismic data. Although this procedure is commonly referred to as “conditioning” the geostatistical model, conditional probabilities do not enter the simulation process and the conditioning process is neither statistical nor probabilistic. As a result, the observations and the modeled results may be substantially different.
In additional, geostatistical methods do not generally distinguish azimuthal variations (in other words, variations eastward from variations westward, northward from southward, or upward from downward). Geologic processes often have such directionality (in other words, beds thinning upward, clay content increasing offshore, and the like), and such variations may be important in determining locations, volumes, and extents of hydrocarbon reservoirs.
Markov-related modeling approaches have been proposed for geostatistical analysis. These approaches do not involve Markov chain analyses however, but merely refer to the well-known Markovian assumption that distant observations can be ignored if closer ones exist along the same direction. This assumption provides limited benefit to the problem of analyzing spatially varying noise in seismic data.
The once popular method of trend-surface analysis, a curve-fitting approach, is no longer commonly applied to geologic data and has had limited application to the field of seismic-data analysis. Trend-surface analysis is similar to geostatistics in that it involves interpolation rather than uncertainty analysis, and therefore offers limited benefit to the spatially varying noise problem.
Signal processing and filtering methods generally involve either an averaging scheme or specification of an a priori error (in other words, noise) model. Typically, the error model is based on a Gaussian distribution with fixed mean and variance. Such techniques are difficult or impossible to apply to systems where the uncertainty may not be Gaussian or where the noise varies spatially in a complex or unpredictable manner.
Two other techniques involving Markov-oriented process are referred to Markov Random Fields and Markov Chain Monte Carlo.
Markov Random Fields is a method for simulating systems that are self-organizing wherein the state of one cell is adjusted, given the states of its neighbor cells, to minimize the energy of the system. The common example of this modeling is for a magnetic material being slowly cooled through its Curie point. The resulting models are not driven by observations of a single embodiment of a physical system, but strive to provide a mathematical description of the class of physical systems. In the magnetic material cooling example, the Markov Random Field does not model any particular magnet, but rather model the behavior of the magnetic material in general. For that reason, the technique is of limited value for studying the noise characteristics of a specific seismic data set.
Markov Chain Monte Carlo is a sampling method for simulating random samples from a multivariate distribution, given the marginal distributions. The Markov chain arises in the process of simulating successive samples, and not in the description of the data itself. Most applications involve developing numerical solutions to complex Bayesian-probability problems.
More specifically, a Markov chain is a probabilistic model describing successive states of a system. Given the present state, the next state in the future does not depend upon the past states, but only upon the present state. Thus, in a Markov chain, the probability of a transition to future state Si+1 from a previous chain of states S1, S2, . . . , Si, is given by
 Pr(Si+1|S1, S2, . . . , Si)=Pr(Si+1|Si),
where Pr(A|B) represents the probability of the occurrence of state A given the existence of the state, or set of states, B.
Ordinary Markov chains describe successive states of a system. For instance, the successive states could be used to describe the time succession of the number of tasks queued for computer processing. Markov chains can also describe successive states of a system in space rather than in time. The descriptions of state successions for a Markov process are encoded in a transition probability matrix. As its name implies, the transition probability matrix contains the probability of going from any specified state to any other state in the next time or location step.
Applications to one-dimensional data dominate the published methods of using Markov chains in the analysis of spatial data for geologic systems. One common example involves the analysis of the lithologic (rock type) succession of a geologic section. However, the exact lithologies must be presumed to be known in geologic models using Markov chains, a limitation of the method.
Markov chain analysis has been shown to be mathematically similar to one form of geostatistics called “indicator Kriging,” and based on that similarity Markov chain analysis has been applied to sparse well data. However, as with geostatistics, this application does not include the capability to evaluate the uncertainty in the observations, which must therefore be presumed to be error-free. The modeling results, therefore, correspond to those produced through geostatistics, and have the same limitations. Thus, neither geostatistics nor ordinary Markov chain analyses allow evaluation of alternative descriptions for noisy data.
A method for characterizing the heterogeneity of subsurface geological formations using an extension of Markov chain analysis has also been proposed. This method is based on the concept of conditioning a Markov chain on future states, and therefore allows conditioning on all available well data. However, as with the methods discussed above this method does not consider the noise inherent to the data itself, and is therefore essentially a means of interpolating sparse data.
Thus, a need exists for a method that can characterize noise in seismic data, allow one to remove that noise and evaluate the uncertainties. Possessing such capability would enhance the reliability of petroleum exploration and development decisions and allow better assessment of decision risks.